On optional stopping of some exponential martingales for Lévy processes with or without reflection

Sören Asmussen and Offer Kella

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
Kella Whitt (1992) introduced a martingale $\{M_t\}$ for processes of the form $Z_t=X_t+Y_t$ where $\{X_t\}$ is a L\'evy process and $Y_t$ satisfies certain
regularity conditions. In particular this provides a martingale for the case where $Y_t=L_t$ where $L_t$ is the local time at zero of the corresponding reflected L\'evy process. In this case $\{M_t\}$ involves, among others, the L\'evy exponent $\varphi(\alpha)$ and $L_t$. In this paper, conditions for optional stopping of $\{M_t\}$ at $\tau$ are given. The conditions depend on the signs of $\alpha$ and $\varphi(\alpha)$. In some cases optional stopping is always permissible. In others,
the conditions involve the well known necessary and sufficient condition for optional stopping of the Wald martingale $\{e^{\alpha X_t-t\phi(\alpha)}\}$, namely that $\tilde{P}(\tau<\infty) $ $=1$ where $\tilde{P}$ corresponds to a suitable exponentially tilted L\'evy process.
Key words:
exponential change of measure, Lévy process, local time, stopping time, Wald martingale